A077177 Number of primitive Pythagorean triangles with perimeter equal to A002110(n), the product of the first n primes.

0, 0, 1, 0, 1, 2, 3, 5, 8, 17, 34, 59, 111, 213, 396, 746, 1413, 2690, 5147, 9826, 18885, 36269, 69952, 134949, 260743, 504636, 978311, 1899832, 3692980, 7190329, 13994206, 27279898, 53195986

Offset

1,6

Comments

A Pythagorean triangle is a right triangle whose edge lengths are all integers; such a triangle is 'primitive' if the lengths are relatively prime.

Equivalently, number of divisors of s=A070826(n) in the range (sqrt(s), sqrt(2s)). More generally, for any positive integer s, the number of primitive Pythagorean triangles with perimeter 2's equals the number of odd unitary divisors of s in the range (sqrt(s), sqrt(2s)). (A divisor d of n is 'unitary' if gcd(d, n/d) = 1.)

References

A. S. Anema, "Pythagorean Triangles with Equal Perimeters", Scripta Mathematica, vol. 15 (1949) p. 89.

Albert H. Beiler, "Recreations in the Theory of Numbers", chapter XIV, "The Eternal Triangle", pp. 131, 132.

F. L. Miksa, "Pythagorean Triangles with Equal Perimeters", Mathematics, vol. 24 (1950), p. 52.

Links

Table of n, a(n) for n=1..33.

Randall L. Rathbun, Equal Perimeter primitive right triangles

Formula

a(n) = A070109(A002110(n)) = A078926(A070826(n)).

Example

a(5) = 1 since there is exactly one primitive Pythagorean triangle with perimeter 2*3*5*7*11; its edge lengths are (132, 1085, 1093). a(7) = 3; the 3 triangles have edge lengths (70941, 214060, 225509), (96460, 195789, 218261) and (142428, 156485, 211597).

Mathematica

a[n_] := Length[Select[Divisors[s=Times@@Prime/@Range[2, n]], s<#^2<2s&]]

Prog

(PARI) semi_peri(p)= {local(q, r, ct, tot); ct=0; tot=0; pt=0; fordiv(p, q, r=p/q-q; if(r<=q&&r>0, print(q, ", ", r, " [", gcd(q, r), "] "); if(gcd(q, r)==1, ct=ct+1; if(q*r%2==0, pt=pt+1; ); ); tot=tot+1); ); print("semiperimeter:"p, " Total sets:", tot, " Coprime:", ct, " Primitive:", pt); } /* Lists all pairs q, r such that the triangle with edge lengths (q^2-r^2, 2qr, q^2+r^2) has semiperimeter p. */

Crossrefs

Cf. A002110, A070109, A070826, A078926.

Sequence in context: A108054 A342690 A123612 * A303874 A145793 A113879

Adjacent sequences: A077174 A077175 A077176 * A077178 A077179 A077180

Keyword

more,nonn

Author

Kermit Rose and Randall L Rathbun, Nov 29 2002

Extensions

Edited by Dean Hickerson, Dec 18 2002

Status

approved